# Subspace Stabilization Analysis for Non-Markovian Open Quantum Systems

## TL;DR

This paper analyzes the stabilization of subspaces in non-Markovian open quantum systems, providing algebraic and Lyapunov-based conditions for invariant and attractive subspaces relevant to quantum information devices.

## Contribution

It introduces analytic conditions for subspace invariance and attractivity in non-Markovian quantum systems, combining algebraic methods with Lyapunov functionals.

## Key findings

- Necessary and sufficient conditions for subspace invariance.
- Sufficient conditions for subspace attractivity.
- Numerical example demonstrating theoretical results.

## Abstract

Studied in this article is non-Markovian open quantum systems parametrized by Hamiltonian H, coupling operator L, and memory kernel function {\gamma}, which is a proper candidate for describing the dynamics of various solid-state quantum information processing devices. We look into the subspace stabilization problem of the system from the perspective of dynamical systems and control. The problem translates itself into finding analytic conditions that characterize invariant and attractive subspaces. Necessary and sufficient conditions are found for subspace invariance based on algebraic computations, and sufficient conditions are derived for subspace attractivity by applying a double integral Lyapunov functional. Mathematical proof is given for those conditions and a numerical example is provided to illustrate the theoretical result.